1
Math 3 Exam 2 Review Guide
These are examples of some types of problems that may appear on the exam. The list is not
exhaustive.
1. Graph the following functions. Show asymptotes where they exist, label maximum and
minimum pairs where applicable, label axes and all “quarter points” with correct scale. Find the
domain and range of the functions. For tangent and cotangent functions, find and label key points
(i.e. the point (π/4, 1) on y = tan x, the point (π/4, 2) on y = 2 tan x, etc.):
c) h(x) = − csc(− π2 x + π4 )
b) f (x) = −5 sin(3x − π2 ) + 1.5
a) y = 4 cos(x − π/6)
e) K(x) = 5 tan( x3 − π4 )
d) g(x) = −2 cot(2x − π) + 1
h) q(x) = sin(−2πx + π2 )
g) w(x) = tan(−2x) − 1.5
f) s(x) = 2 sec(3x) + 2
M (x) = − 41 sec(x + π2 ) − 2
2. On the function y = −3 csc(−πx − 32 ) − 15 , find:
a) A b) Period c) Phase shift d) Vertical shift e) Vertical reflection f) Horizontal reflection
g) Find all asymptotes.
3. Prove the following identities:
a)
sin2 α
tan4 α
3 csc3 α
cot6 α
2
=1
b)
cos γ
1−tan γ
+
sin γ
1−cot γ
= cos γ + sin γ
c) sin 8θ = 8 sin θ cos θ(1 − 2 sin2 θ)(1 − 8 sin2 θ cos2 θ)
d)
cot(−t)+csc(−t)
sin(−t)
e) cos θ +
f)
1
tan α+tan β
π
4
=
=
1
1−cos t
√
=
2
(sin θ
2
+ cos θ)
cos α cos β
sin(α+β)
g) cos4 x − sin4 x = cos 2x
h)
sin u−sin v
cos u−cos v
= − cot 12 (u + v)
4. a) Find all points where the graph of y = csc5 x crosses the graph of y = 4 csc x
b) Find all points where the graph of y = 2 cos3 t + cos2 t − 1 intersects the graph of y = 2 cos t
2
5. Find all solutions of the following equations:
a) sin β + 2 cos2 β = 1
b)
√
3 tan 5θ − 1 = 0
c) tan 2x cos 2x = sin 2x
d) tan 12 θ = csc θ − cot θ
e) 2| sin 3β| −
√
√
3 = −2 3
f) cos 5x cos 3x =
1
2
+ sin(−5x) sin 3x
g) sin x − sin 2x = − sin 3x
h) 2 sin 3x cos 3x =
6. If
π
2
< θ < π and
a) sin(α − θ)
1
2
π
2
< α < π, given csc θ =
b) cos(α + θ)
5
3
8
and cos α = − 17
find:
c) tan(θ − α)
d) tan 2θ
e) cos 12 θ
7. In the following equation y describes the outdoor temperature in ◦ F on a non-existent island
during a 24-hr period and t describes the time in hours, with t = 0 corresponding to 9am. Find
the lowest outdoor
and what time it occurs.
π temperature
(t − 3)
y = 80 + 22 cos 12
π
and travels at a rate of 300ft/sec. Approximately how
8. An airplane takes off at an angle of 18
long will it take the airplane to reach an altitude of 15,000 ft? Give the time in minutes, rounded
to the nearest minute.
9. Reference Problem 33 drawing from the text:
The Tower Bridge in London consists of two towers, each connected to either side of the Thames
river, with a distance between them of 200ft. Each “leaf” (or side) of the drawbridge can be
raised to an angle of elevation of 83◦ . If the water level is 15ft below the closed bridge during a
particular tidal stage, Approximate the distance d, to the nearest tenth of a foot, between the end
of a drawbridge leaf and the water level if the bridge is fully opened at the same tidal stage. Can a
ship with a maximum height of 99ft and maximum width of 75ft pass under the drawbridge when
it is fully opened?
10. A rectangular box has dimensions 8in x 14in x 6in. Approximate, to the nearest inch, the
diagonal length between a lower corner and an upper corner of the box. Also approximate, to the
3
nearest second, the angle formed by the diagonal and a diagonal of the base, as shown.
11. A sheer vertical cliff has a height of 100m. A person standing on the top of the cliff is
holding a camera 1m above the ground. They calculate the angle of depression from the camera
lens to a seal on the beach below is 30◦ . Find the exact distance from the camera to the seal, and
the exact distance from the seal to the base of the cliff.
12. Find the exact values of the following expressions:
a) cos( 3π
− π3 )
4
b) tan(105◦ )
c) sin 2θ (if csc θ = − 53 and 0◦ < θ < 270◦ )
d) tan π8
e) sin 165◦ + sin 105◦
f) sin 285◦
g) cos(− 19π
)−
12
h) sin(α + β) and the quadrant that α + β lies in (if sin α = − 45 , sec β =
is in QIV)
13. a) Express as a Product: sin 8t + sin 2t
b) Express as a Sum or Difference: 2 sin 7θ sin 5θ
c) Express in terms of a cosine function with exponent 1: sin4
14. a) If tan 2θ =
1
2
and π < θ <
3π
4
x
2
find the exact value of tan θ.
b) Find the exact value of sin 2x if sin x =
15. Find an equation for the given graph.
3
4
and
π
2
< θ < π.
13
,
12
√
6
4
α is in QIII, and β
4